If the amounts of two consecutive years on a sum of money are in the ratio 20 : 21, find the rate of interest.

Answer

To solve the problem of finding the rate of interest when the amounts of two consecutive years are in the ratio 20:21, we can follow these steps:

### Step 1: Understand the problem
We know that the amounts after two consecutive years are in the ratio of 20 to 21. Let's denote:
- Amount after the first year = A1
- Amount after the second year = A2

Given that:
\[ \frac{A1}{A2} = \frac{20}{21} \]

### Step 2: Express the amounts in terms of the principal and interest
Using the formula for compound interest, we can express the amounts:
- Amount after the first year (A1) can be expressed as:
\[ A1 = P \left(1 + \frac{R}{100}\right) \]
- Amount after the second year (A2) can be expressed as:
\[ A2 = P \left(1 + \frac{R}{100}\right)^2 \]

### Step 3: Set up the ratio
From the ratio given:
\[ \frac{A1}{A2} = \frac{20}{21} \]

Substituting the expressions for A1 and A2:
\[ \frac{P \left(1 + \frac{R}{100}\right)}{P \left(1 + \frac{R}{100}\right)^2} = \frac{20}{21} \]

### Step 4: Simplify the equation
The principal (P) cancels out:
\[ \frac{1 + \frac{R}{100}}{\left(1 + \frac{R}{100}\right)^2} = \frac{20}{21} \]

This simplifies to:
\[ \frac{1}{1 + \frac{R}{100}} = \frac{20}{21} \]

### Step 5: Cross-multiply to solve for R
Cross-multiplying gives:
\[ 21 = 20 \left(1 + \frac{R}{100}\right) \]

Expanding this:
\[ 21 = 20 + 20 \cdot \frac{R}{100} \]

### Step 6: Isolate R
Subtract 20 from both sides:
\[ 1 = 20 \cdot \frac{R}{100} \]

Now, multiply both sides by 100:
\[ 100 = 20R \]

Divide both sides by 20:
\[ R = \frac{100}{20} = 5 \]

### Conclusion
The rate of interest (R) is 5%.

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